JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,
A three-dimensional model of wave attenuation in the marginal
ice zone
L. G. Bennetts1, M. A. Peter2, V. A. Squire1 and M. H. Meylan3.
Abstract.
A three-dimensional model of wave scattering by a large array of floating thin-elastic
plates is used to predict the rate of ocean wave attenuation in the marginal ice zone in
terms of the properties of the ice cover and the incoming wave field. This is regarded
as a small step towards assimilating interactions of ocean waves with areas of sea ice into
oceanic general circulation models.
Numerical results confirm previous findings that attenuation is predominantly affected
by wave period and by the average thickness of the ice cover. It is found that the shape
and distribution of the floes and the inclusion of an Archimedean draft has little impact
on the attenuation produced. The model demonstrates a linear relationship between ice
cover concentration and attenuation. An additional study is conducted into the directional evolvement of the wave field, where collimation and spreading can both occur depending on the physical circumstances. Finally, the attenuation predicted by the new
three-dimensional model is compared with an existing two-dimensional model and with
two sets of experimental data, with the latter producing convincing agreement.
the sea ice and collisions between nearby floes [Squire et al.,
1995]. An intimate relationship between the configuration
of the ice cover in the MIZ and the traveling waves arises,
whereby the ice cover removes energy from the waves, whilst
the waves determine the morphology of the ice cover. Evidence of this is seen in the formation of floes into zones of
increasing diameter, ordered in distance away from the ice
edge [Squire and Moore, 1980].
Although rising temperatures are believed to be the primary threat to sea ice, it is conjectured that its effect is
compounded by ocean waves. Direct wave-induced melting
has been confirmed as significant by Wadhams et al. [1979],
especially in the outer regions of the MIZ. However, penetrating ocean waves also act to break up floes allowing increased contact between local open water and ice, which will
hasten the annihilation of sea ice indirectly under summer
conditions at least. Disturbing the balance between waves
and sea ice with a reduction in the strength, compaction
and extent of the ice cover, leads to increased wave activity
and a corresponding amplification of these direct and indirect destructive agents, thus defining a positive feedback
loop. In addition to these climatological implications, the
presence of waves in regions of sea ice is a consideration for
the engineering activities that take place in its vicinity.
Whilst sea ice has been integrated into many contemporary global climate models, at present there exists no mechanism for assimilating the influence of ocean waves on the
sea ice. The purpose of the work presented herein is to study
the evolution of long-crested ocean waves of a prescribed period through the MIZ, given a snapshot of the prevailing ice
conditions. In this sense it can be regarded as a small step
towards providing a coupled waves and sea ice component
for use in an oceanic general circulation model.
The ability to describe the propagation of waves that
travel through regions of ice-covered fluid has received a
considerable amount of attention in the last twenty years.
From the highly idealized early models, the science has now
advanced to the stage at which properties such as ice of
varying thickness and a correct Archimedean draft can be
accommodated [Bennetts et al., 2007, 2009] and the consideration of scattering by three-dimensional floes of arbitrary
shape is possible [Meylan, 2002]. A summary of the recent
advances in the theory of waves and sea ice is presented by
1. Introduction
Immense regions of sea ice encircle the Antarctic continent, stretching far into the Southern Ocean and seasonally
expanding and contracting the cryosphere. The existence of
this natural barrier is fundamental to the survival of the ice
shelves and ice tongues that abound the continent’s coastline [Squire et al., 1994; Cathles et al., 2009]. But sea ice is
also an immensely geophysically significant substance itself,
which, for example, influences marine ecology and affects the
global climate system. The latter contribution is largely due
to the summer ice albedo feedback mechanism [Hall , 2004],
but other factors, such as the production of dense water during freezing and the expulsion of fresh water during melting
are also important [Feltham, 2008].
The ocean wave motion that most affects sea ice, the topic
of this work, is confined to a dynamic region known as the
marginal ice zone (MIZ) that reaches for tens to hundreds
of kilometers from the ice margin into the ice pack [Squire
et al., 1995]. Thereabout the ocean surface is a mixture of
sea ice and open water, where the ice consists of many different floes of varying shapes and sizes that are free to move
their position within the mélange when acted upon by winds
and waves.
The proportion of a wave that withstands reflection at
the ice edge and is transmitted into the pack causes the ice
floes to flex. In doing so, a strain is imposed on the ice
that may weaken it and, if large enough, cause fracture and
subsequent break up [Langhorne et al., 2001]. As the wave
progresses deeper into the ice pack it is attenuated through
diffraction that arises as it meets each floe in its path, and
additionally by other natural agents such as hysteresis in
1 Department of Mathematics and Statistics, University of
Otago, Dunedin, Otago, NZ.
2 Institute of Mathematics, University of Augsburg,
Germany.
3 Department of Mathematics, University of Auckland,
Auckland, NZ
Copyright 2010 by the American Geophysical Union.
0148-0227/10/$9.00
1
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
Squire [2007]. However, assumptions of linear motions and
harmonic time-dependence remain a feature amongst these
works and the concept of treating the sea ice as a thin-elastic
plate is universal.
The next step is to reconstruct a region of sea ice of realistic proportions by combining a large number of scattering sources and this presents its own significant challenges.
Dixon and Squire [2001] attempt this using a coherent potential approximation. However, the investigation was confined to two dimensions and a three-dimensional extension
is thus far lacking.
Efforts to model wave propagation in the MIZ have also
been made in the past by simplifying interactions so that
they are in terms of energy alone. The first attempt of this
kind was made by Wadhams [1986] for a two-dimensional
geometry. This was extended to three-dimensions by Masson and LeBlond [1989], and non-linear wave coupling, dissipation of wave energy and wind-wave generation were accounted for, notwithstanding the assumption of isotropy of
the ice cover in order to generate numerical solutions. In
an ensuing work, Perrie and Hue [1996] combined the approach of Masson and LeBlond [1989] with an operational
wave model to predict wave attenuation in the MIZ. However, although they included three degrees of motion and
draft, both of these works were restricted by the need to
model the floes as rigid circular bodies. Flexure of the ice
was accommodated by Meylan et al. [1997] in a zero-draft
floe model based on a linear Boltzmann (or transport) equation, which has recently been shown by Meylan and Masson
[2006] to be almost identical to the multiple scattering theory of Masson and LeBlond [1989].
In the energy scattering descriptions developed in the
above papers, wave interactions take place without the consideration of phase effects. However, evidence now suggests that the accurate calculation of average attenuation
is highly dependent on the inclusion of such features [Berry
and Klein, 1997]. Although relevant theories exist for interactions that acknowledge the phase of waves [for example
Peter and Meylan, 2004], in practice a direct application to
an ice covering on the scale we wish to investigate is a massive and fanciful numerical undertaking. For this reason,
previous investigations, notably Kohout and Meylan [2008],
Vaughan et al. [2009] and Squire et al. [2009], have been restricted to homogeneity in one horizontal dimension. Whilst
the latter two studies were primarily concerned with wave
evolution in the Arctic Basin, where the sea ice forms a
quasi-continuous veneer, the MIZ is composed of many separate floes and it is widely accepted that to represent it
faithfully demands a fully three-dimensional theory. In this
sense the current work can be viewed as an extension of
Kohout and Meylan [2008].
Recently, two similar but independent mathematical
methods have been developed by the authors for modeling
wave scattering in the MIZ (Bennetts and Squire, 2009a, b,
and Peter and Meylan, 2009a, b). These models treat the
MIZ as a three-dimensional expanse assembled from a vast
collection of floes that are simultaneously individual and interrelated. In the models the MIZ is idealized as an array
that is fashioned from a large number of rows, where each
row consists of an infinite number of floes, and it is possible
to vary the properties of the floes and the various spacings
involved. To combine the motions of the floes in a row, a
periodicity condition is applied and a common periodicity
is required of the array as a whole to harmonize the row
interactions.
In the current work we will combine the two methods and
fulfill the role that the underlying model was originally conceived to replicate, as a basis for investigating attenuation
of long-crested ocean waves in the MIZ due to diffraction by
floes. The availability of two different methods that agree
so well, for such a complicated situation, gives us great confidence in our output but their relative strengths also complement one another and will allow us to examine how characteristics of the MIZ, such as floe shape and concentration,
determine the degree of attenuation produced.
The following section begins by outlining the threedimensional mathematical model of the MIZ and giving a
brief overview of the two solution methods. At the end of
§2, the process of extracting an attenuation coefficient, and,
in particular, the use of ensemble averaging to eliminate obstructive coherence effects, is discussed. Subsequently, in §3
we conduct a numerical investigation into the dependence of
the attenuation coefficient on several key properties of the
MIZ, focusing on those that are new to our model, such as
concentration, draft, floe shape and directionality. The section finishes by contrasting the attenuation predicted by our
three-dimensional model with that of the two-dimensional
model of Kohout and Meylan [2008]. In the penultimate
section, two sets of experimental data on wave attenuation
in the MIZ are chosen for comparison with our model and
a generally creditable agreement is found. To conclude, a
summary of the work presented in this article is given in §5.
2. The 3-dimensional mathematical model
2.1. Preliminaries
We visualize the MIZ as an array consisting of a large
number of sea-ice floes floating on an otherwise open fluid
domain of finite depth h that is infinite in all horizontal directions. The principal attribute of the physical situation we
wish to investigate is the attenuation of ocean waves due to
the scattering produced by the floes. We therefore disregard
processes such as viscosity and floe collisions. The region of
the horizontal plane occupied by the ice cover is considered
fixed and the surge response of the floes is neglected.
The array is composed of a series of rows, in which each
row contains an infinite number of modules of floes with
some prescribed periodicity imposed to facilitate a solution.
For the majority of our investigation the modules will contain a single floe only, so that the rows consist of identical
floes. However, in §3.3 the effects of including an additional
floe in the modules, which allows for a non-uniform distribution of floes in the rows, is studied. The dimensions of the
ice cover are assumed to be known but will be randomized in
order to simulate the natural heterogeneity of the situation.
Its structural properties are also considered known with its
flexural rigidity given by F = Y D3 /12(1 − ν 2 ), where we
set the Young’s modulus for sea ice as Y = 6 × 109 Pa and
Poisson’s ratio as ν = 0.3, these being typical values, and D
denotes the ice thickness. Each floe in the array is free to
flex independently of its neighbors but its motion is affected
by the waves diffracted away from all of the other floes. The
geometry just described is depicted in figures 1–2.
In keeping with other similar studies, we suppose that the
waves passing through the MIZ are of small amplitude, so
that linear theory may be applied, and consider solutions at
a given period τ s. Under these conditions, and assuming
an irrotational velocity field, the motion of the fluid may
be described using a velocity potential Φ = Φ(x, y, z). The
coordinates x and y determine the position in the horizontal
plane, with y orientated so that it lies parallel to the rows,
and z is the vertical coordinate that points directly upwards
and has its origin set to coincide with the fluid surface in
the absence of ice (see figure 1). Treating the fluid as incompressible and inviscid, the velocity potential, Φ, is governed
throughout the fluid domain by Laplace’s equation
∇2 Φ = 0
(x, y ∈ R2 ),
for −h < z < 0 when ice does not occupy the fluid surface and for −h < z < −d when ice is present, where
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
d = (ρi /ρw )D is the draft of the floes, in which ρi = 922.5 kg
m−3 is the density of sea ice and ρw = 1025 kg m−3 is the
density of the underlying water. On the ocean bed, z = −h,
the no-flow condition Φz = 0 holds, and on the linearized
fluid surface away from the floes, z = 0, we have the freesurface condition Φz = σΦ, where the frequency parameter
is σ = (1/g)(2π/τ )2 and g ≈ 9.81 m s−2 is acceleration due
to gravity.
Sea ice forms floes that have horizontal dimensions that
far exceed their thicknesses, so that fluid motion will induce
a flexural response [Squire et al., 1995]. Combined with the
assumption of small amplitude waves, this fact motivates
the modeling of sea ice as a thin-elastic plate, for which
points in a vertical plane retain their alignment under deformation [Timoshenko and Woinowsky-Krieger , 1959]. For
a thin-elastic plate it is possible to calculate the stresses
and strains experienced by the ice under fluid motion from
knowledge of the displacement it undergoes on its lower surface, denoted W = W (x, y).
At the fluid-ice interface the velocity potential and displacement function are linked by relating the thin-elastic
plate equation, which describes the motion of the ice in
terms of the difference in pressure at its lower surface and the
(constant) atmospheric pressure, to the linearized version of
Bernoulli’s equation, which provides an expression for the
fluid pressure beneath the ice. This assumes no cavitation
between the ice and the fluid. Adding a linearized kinematic
condition to these equations results in the coupling
has the form
Φ(x, y, z) ∼
X
Tm eik(vm x+um y) cosh{k(z+h)}
as x → ∞,
m∈M
(2)
where vm = cos χm , um = sin χm and the Tm are amplitudes that must be calculated [Peter et al., 2006; Bennetts
and Squire, 2009a]. The size of the set of transmitted wave
angles varies according to a number of parameters. Typically, in the simulations we are interested in, it consists of
only a handful of values (≤ 5) and often merely the incident
wave angle. An investigation of the directional spectrum is
made in §3.1.
In order to produce the results for our investigation, two
solution methods for the geometrical situation described
above are utilized. Possessing two such solution methods,
which have been formulated independently and employ different mathematical techniques, has allowed us to validate
thoroughly the results that we will present. Moreover, although the two methods solve essentially the same problem,
each one has its own particular advantages that will allow
us to probe certain features of the ice cover.
z
d
ice
D
(1 − σd)W + (F/ρw g)(Wxx + Wyy )2 − Φ = 0,
which is applied at the linearized fluid-ice interface z = −d.
There are also so-called free-edge conditions, which impose
the vanishing of the ice’s bending moment and shearing
stress, and must be enforced at the perimeter of each floe.
These are represented through the equations
Wxx + Wyy − (1 − ν) (Wss + ΘWn ) = 0,
(1a)
(Wxx + Wyy )n + (1 − ν)Wsns = 0,
(1b)
and
water
h
bed
Figure 1. Figure showing the geometry in the vertical
plane including a typical floe.
respectively, and are the natural conditions that occur when
Hamilton’s principle is applied to the thin-elastic plate equations [Porter and Porter , 2004]. In equations (1a-b) the subscript n denotes differentiation with respect to the normal
direction and s the tangential direction, when traversing the
perimeter of a floe. The quantity Θ denotes curvature. One
final condition must hold at the edges of the floes, which
describes the fluid’s inability to penetrate through the submerged portion of the floe, and is
Φn = 0
(−d < z < 0).
A long-crested ocean wave, ΦI , is incident on the array
of floes and travels from the far-field x → −∞ at an oblique
angle χ with respect to the x-axis. This incident wave may
be expressed as
Figure 2. Figure showing the geometry in the horizontal
plane. In this case the modules contain four floes.
ΦI (x, y, z) = eik(v0 x+u0 y) cosh{k(z + h)},
where v0 = cos χ, u0 = sin χ and k is the propagating
wavenumber, which is defined as the real, positive root of
the dispersion relation
k tanh(kh) = σ.
During the scattering process, the array of floes redistributes
the energy carried by this wave over a finite number of angles χm (m ∈ M ⊂ Z), so that the transmitted wave field
Details of these methods are described by Bennetts and
Squire [2009a, b] and Peter and Meylan [2009a, b] respectively. These previous works formulated the solution procedures and investigated the main mathematical and computational properties contained therein. In contrast, the current work focuses on the physical implications that can be
drawn from using this three-dimensional model. To achieve
this, we will make use of the key findings presented by Bennetts and Squire [2009a, b] and Peter and Meylan [2009a, b],
which are outlined in the following two subsections.
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
2.2. Row spacing
Between the rows of ice floes both propagating and
evanescent waves exist. However, for numerical expediency,
in the current work it is assumed that the interactions consist only of the former, with the latter considered negligible
during the process. Such a restriction is well established
in many areas of wave interactions, for instance hydrodynamic problems [see Linton and McIver , 2001] and electromagnetic scattering [for example McPhedran et al., 1999],
and in the former it is often termed a wide-spacing approximation (WSA). Although it is possible to incorporate the
effects of the evanescent waves in the interactions, it was
shown by both Bennetts and Squire [2009b] and Peter and
Meylan [2009a] that the restriction to propagating waves
only is of high accuracy and numerically efficient.
Moreover, it has been mentioned already that our model
only accounts for attenuation caused by scattering, with
waves propagating unhindered through open water. Thus
the use of the WSA makes it explicit that the spacing between rows will only affect the phase of the waves traveling
between them. Any resulting modulation of the transmitted energy will be eradicated by sufficient averaging and
thus row spacing ceases to be a determinant of the attenuation properties. This has no implications for our attenuation coefficient, α, which will be defined in the following
section, due to the non-dimensionalization employed but it
will have important ramifications for the concept of concentration that is discussed in §3.4.
2.3. Attenuation
It is well established through experimental evidence [e.g.
Wadhams, 1975, 1978; Squire and Moore, 1980] that wave
energy attenuates exponentially through the MIZ and that
this attenuation acts as a low-pass filter that favors the
transmission of long waves. Exponential attenuation is also
a property of linear models such as the one employed in this
work [Kohout and Meylan, 2008]. Our investigation will
center on the degree of attenuation arising from certain key
attributes of the ice cover, such as its concentration on the
ocean surface and ice thickness.
In order to study wave attenuation we define the transmitted energy E to be
E=
resonate over the length of a row and a row separation if
this scale occurs too many times [so-called Bragg resonance,
see Bennetts and Squire, 2009b] and this will produce an artificially large attenuation over a wide range of periods. It is
also possible for resonances emanating from individual rows
to contaminate results unless care is taken in the calculation
of the attenuation coefficient.
An established method to avoid these inhibiting resonance effects is to perform ensemble averaging [see Kohout
and Meylan, 2008]. Specifically, we construct E as a function of Λ by taking the mean of the transmitted energies
given by a large number of arrays, typically one hundred,
in which certain key parameters are distributed normally
around given values. The exponential decay, described in
equation (3), of this averaged function is then calculated
using a least squares approach to yield the attenuation coefficient α. In the results section the particular methods of
averaging used will be described in detail and their relation
to the type of resonance that necessitates their implementation.
3. Simulation of attenuation in the MIZ
In this section the relative influence of the various parameters that exist in the problem on the rate of attenuation will
be investigated. In doing so we wish to extend the study of
Kohout and Meylan [2008]. Using a two-dimensional model
of the MIZ, Kohout and Meylan [2008] found that the rate
of attenuation is primarily determined by ice thickness and
wave period. Clearly thickness and period will also be important parameters in our model. However, in our threedimensional model there are many additional features that
may be of importance to the passage of waves through an
ice-covered ocean can be investigated.
3.1. Directionality
We begin by investigating the effect of the angle of the
incident wave on attenuation. Recall that in our model it is
possible to set the direction at which the crest of the incident wave impacts on the array by means of the parameter
v0 = cos χ, which ranges from normal incidence v0 = 1 to
grazing incidence v0 = 0. This then modifies the angles at
1 X
|Tm |2 vm ,
v0 m∈M
which is the wave energy traveling in the direction of the incident wave field, taken over the discrete spectrum of angles
generated by the array of floes. The value of this quantity will depend on both the properties of the incident wave
field and the ice cover, in particular the number of rows, Λ
say, comprising the array. As described above, we expect E
to decay exponentially with distance through the MIZ, and
thus, following Kohout and Meylan [2008], we may separate
the dependence of the transmitted energy on row number
by writing
E = e−αΛ ,
(3)
−1
10
α
(a)
−2
10
−3
10
−4
10
−2
10
α
(b)
−3
10
−4
10
−5
10
−3
10
(c)
−4
α
10
−5
10
where α is referred to as the (non-dimensional) attenuation
coefficient.
The primary objective of this work is to determine the behavior of the attenuation coefficient, as parameters such as
ice thickness and wave period are varied, and also to compare these results to those already existing for simplified,
two-dimensional models of the MIZ and experimental data.
This requires us to extract an attenuation coefficient from
our three-dimensional model. The process that is required
to achieve this is complicated by the occurrence of constructive and destructive coherence. For instance, waves may
−6
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
cos χ
Figure 3. The attenuation coefficient against the cosine
of the incident angle for an array of circular floes that
have a 10 m radius and are 0.5 m thick (solid lines) or 2 m
thick (broken lines), with straight-line fits superimposed
(thick lines). The in-row spacing is 5 m and submergence
is neglected. Part (a) shows the case of 8 s waves, part
(b) 12 s waves and part (c) 16 s waves.
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
←
0.9
7
8
9
−45◦
← 16 s
12 s
0.8
0.7
0.5
0.4
←
0.3
8s
0.2
0.1
−60
−30
0
0
angle
45◦
Figure 5. Vectors depicting the components of the energy transmitted by an array composed of one hundred
rows. The floes used in the array have mean thickness 1 m
and diameter 100 m and the in-row separation is 50 m.
The direction of the vectors denotes the angle of the
transmitted waves, with the vertical set as being normal to the array, and their magnitudes are derived from
the value of the energy carried by the particular wave.
0.6
0
−90
6
10
1
E
percent of the central wave packet remains and the qualitative nature of the incident wave is undisturbed. Attenuation of the 8 s wave, however, is far more pronounced. For
instance, only around half of the incoming wave energy traveling straight through the array still exists. Furthermore, as
predicted by the previous figure, the attenuation of the energy increases with incident angle and no discernible energy
remains traveling beyond a 65◦ angle.
From the investigation conducted in this section thus far,
we therefore deduce an exponential dependence of the attenuation on the direction of the incident wave, with its rate
highly dependent on the wave period. Having established
this, unless otherwise stated, for simplicity we will proceed
using only normal incidence (v0 = 1).
There is another interesting aspect to the question of directionality in our three-dimensional model. It was mentioned earlier that, as well as directly attenuating the amplitude of the incident wave, the array of floes may transmit waves that travel in other directions (cf. equation 2).
However, supplementary waves will only be generated under
certain conditions. In particular, large floes and long in-row
separations allow more scope for auxiliary waves to be created and, often in simulations, where the arrays are tightly
packed and consist of relatively small floes, no such waves
will be seen. Nevertheless, for a given geometry, if small
enough periods are considered then extra waves are always
produced, and for some cases the period range in which this
takes place is physically admissible.
period (s)
which waves travel within the array and are ultimately transmitted (see equation 2). Note that, because we measure
attenuation normalized with respect to the incident angle,
the change in the distance that must be traveled by waves
through the ice pack as v0 is varied will not be reflected in
the value of α.
Figure 3 displays the attenuation coefficient α as a function of the cosine of incident angle, v0 , for three different
wave periods τ = 8 s, 12 s and 16 s, which correspond to
the incident wavelengths of approximately 100 m, 225 m and
400 m, respectively. The floes that form the array are all circular, of radius 10 m and with an in-row spacing, defined as
the distance between the closest points of adjacent floes, of
5 m. Draft, which is investigated in the next section, is neglected. In each subfigure the results for two ice thicknesses
D = 0.5 m and 2 m are given and a least-square straight-line
fit is overlayed on each data set.
Note that the ordinate axes here are on a log scale. It
is apparent from these data that there is a nearly exponential dependence of the attenuation coefficient on the incident
angle, with normal incidence (at the right-hand end of the
figures) experiencing the least attenuation. However, the
straight-line fits do not match the data so well as the angle
of incidence becomes close to parallel with the array and this
feature becomes more pronounced for longer waves. This is
due to resonant behavior at grazing incidence when all of
the incoming wave is reflected [see Linton and Thompson,
2007].
The influence of the incident angle on the attenuation
rate is greatest at lower periods. Here we see a difference
of more than an order of magnitude between the two ends
of the range of angles for the 8 s waves, which decreases to
approximately half an order of magnitude for 16 s waves. It
is also striking that the ice thickness has negligible influence
over the functional behavior of α with v0 – the corresponding
curves lying virtually parallel to one another.
To give a more visual appreciation of the way in which
the direction of an incident wave is modified as it travels
though the MIZ, figure 4 shows an evolved energy spectrum
for the arrays considered in figure 3 with 2 m ice thickness.
In this figure the transmitted energy is given as a function
of incident wave angle after three hundred rows, which corresponds to approximately a ten-kilometer penetration into
ice-covered water. The incoming wave energy is taken as
a cosine-squared distribution around normal incidence and
this is also shown.
30
60
90
angle
Figure 4. The transmitted energy E as a function of
angle after 300 rows. The row geometry is identical to
that in figure 3. Wave periods 8 s, 12 s and 16 s are shown
(solid lines) along with the energy distribution of the incident wave field (dotted).
Even after this distance into the MIZ the energy transmitted in the 16 s case is barely distinguishable from that of
the incident wave. In the 12 s case the wave energy has experienced attenuation but, despite this, approximately ninety
In figure 5 we consider a feasible situation in which multiple waves are transmitted by the array. Here the array is
composed of one hundred rows of floes of diameter 100 m
and mean thickness 1 m, with an in-row separation of 50 m.
Through the use of vectors the figure depicts the cut-in of
additional waves as period decreases, and the subsequent
development of their properties. The direction of the vectors themselves indicates the angles of the waves and their
magnitudes denote the proportion the corresponding waves
contributes to the transmitted energy E, which is scaled
logarithmically.
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
For the range of periods used in figure 5 there are two instances of waves cutting in, at approximately 8.9 s and 6.2 s.
The extra waves occur in pairs, with each of equal magnitude and directionally symmetric about the incident wave
angle. It is clear that as waves cut in they do so at grazing
incidence and are only of negligible magnitude. The angles
then evolve smoothly, approaching that of the incident wave,
and they contribute an increasingly large proportion of the
overall traveling energy. This quantity is therefore continuous at the points for which the number of propagating waves
changes, although, as we will see, it does change rapidly over
the surrounding intervals.
Experimental evidence suggests that the directional wave
field broadens within the MIZ [Wadhams et al., 1986]. Moreover, isotropy can be attained close to the ice edge, over a
distance as short as a kilometer, but this is heavily dependent on the salient wave period, with long periods only displaying such tendencies far further into the ice pack. The
cause of a directional widening has been attributed to lateral scattering effects [Squire et al., 1995], which compete
with collimating effects that, in contrast, act to narrow the
wave spectrum. The latter can be discerned, although only
slightly, in figure 4.
Due to the restraints they impose on the geometry, twodimensional models are only able to simulate collimation,
whereas, in our three-dimensional model, floes are free to
scatter waves in all directions and thus both features are
accommodated. Whilst we are careful not to assert that our
model exactly replicates the physical phenomenon of wave
spreading, it is nonetheless striking that it does account for
a widening of the transmitted wave field at lower periods.
However, the discrete manner in which the supplementary
waves appear may be somewhat artificial in the context of
wave scattering in the MIZ and we will seek methods to
refine this attribute when developing future models.
3.2. Draft
We now turn to the effects caused by the introduction of
draft into the model so that the floes are neutrally buoyant.
This requires that the floes obey the Archimedean principle and, for the relative values of our chosen ice and water
densities, the draft for each floe must therefore be ninety
percent of its thickness, that is d = 0.9D.
The addition of draft to a mathematical model of an ice
sheet and an ice floe has been studied previously [see Bennetts et al., 2007, 2009; Williams and Squire, 2008; Williams
and Porter , 2009] and it has been shown that it may significantly increase the amount of wave scattering produced,
particularly for prominent features such as large pressure
ridges, although the effects arising for a floe are more subtle. However, the inclusion of draft, in conjunction with
flexure, in a realistic ice field comprising a vast number of
interacting floes has not been conducted previously.
For a range of ice thicknesses, figure 6 compares the attenuation coefficients produced by arrays in which draft is
neglected and arrays in which draft is accommodated. In all
other respects the arrays are identical and they are made up
from circular floes of radius 10 m with an in-row separation
of 5 m. The three wave periods 8 s, 12 s and 16 s are shown
by the different subfigures. In addition to averaging the distance between the rows, in these results the ice thickness
has also been varied using a normal distribution around the
given mean value in order to eliminate length resonances.
−1
10
α
(a)
−3
10
−5
10
−2
10
α
(b)
−4
10
−6
10
−3
10
α
(c)
−5
10
−7
10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Thickness (m)
Figure 6. The attenuation coefficient α as a function
of mean floe thickness comparing ice cover in which an
Archimedean draft is accommodated (solid curves) with
ice cover that uses zero draft (broken curves). The three
subplots show the periods (a) 8 s, (b) 12 s and (c) 16 s.
The row geometry is identical to that in figure 3.
As would be expected, it is clear that the general trend
is for the addition of a physically admissible draft to cause
greater attenuation. Moreover, in most cases this feature is
accentuated by thicker ice. However, the relationship between the corresponding Archimedean draft and zero draft
curves is not always so simple. In particular we note that
for an 8 s period and mean thicknesses beyond approximately 3.5 m the array of zero draft floes attenuates an
equal amount or slightly more than the corresponding array of Archimedean draft floes. This is a complicated case
though, in which much scattering is produced by both arrays and the influence of the submergence can be perceived
to be relatively small.
In the results presented here, the addition of submergence
never causes the attenuation coefficient to deviate as much
as an order of magnitude from the zero draft model. Those
situations in which it does affect the attenuation are for
low periods with thin ice and particularly for the mid-range
period with thick ice, and we note the difference in the corresponding curves for thicknesses up to 2 m for 8 s periods
and thicknesses greater than 1.5 m for 12 s periods. These
intervals contrast with the results for 16 s periods, for which
scattering is minimal so that the introduction of draft is
inconsequential and, as a result, the attenuation is only affected marginally.
At this point we could conclude that it is not essential
to include draft in our calculations in order to predict attenuation accurately. However, the introduction of draft
has an important property, which is of a computational nature and is not highlighted in figure 6. It is known for
two-dimensional models that submergence eliminates the
occurrence of perfect transmission [Williams and Porter ,
2009] and can not be attributed to a length scale [Vaughan
et al., 2007]. This phenomenon is also true in our threedimensional model and numerical tests have shown that the
feature is displaced into another regime. An investigation of
the causes of this is technical and not aligned with the aim
of this paper and will be explored in future work. However,
maintaining draft in our model does have the important
computational benefit of reducing the extent of averaging
X-7
BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
that must be performed to calculate an attenuation coefficient that is free of resonances caused by an overly stylized
geometry.
length of the two floes, so that unity refers to identical bodies, which is the setting used elsewhere in the paper.
3.3. Floe shape and distribution
Until this point in our investigation the arrays that we
have used have all been composed of circular floes of an
identical mean radius. This homogeneity contrasts strongly
with the dynamic and heterogeneous nature of the physical
phenomenon it is trying to describe. Although we expect
that an average property of the array, such as the attenuation it engenders, will be influenced by global properties,
such as concentration (investigated in the following section),
no justification for this assumption currently exists. In this
section we therefore investigate the effect of introducing inhomogeneity to the ice cover in the form of non-circular floe
shape and a non-uniform distribution of the ice cover.
To demonstrate the influence of floe shape, we compare
arrays of square floes and rows of circular floes. For pertinent geometrical parameter sets it is necessary to use a
relatively short incident wavelength for differences to become apparent. An example is given in figure 7, where we
show the wave fields for two rows of alike circles and squares
when the wave period is 6 s and the incident angle is 30◦ .
Despite this short wave exposing the particular scattering
properties of the floe geometries, when the arrays are extended the attenuation coefficients produced are very close,
with α = 0.0378 for an array of circles and α = 0.0535 for
an array of squares. We remark that the wave fields for an
8 s incident wave (not shown) are visually indistinguishable.
Floe 1
Floe 2
rL
L
Figure 8. An example of a module that is used to obtain
the results presented in figure 9. This particular module
contains two circular floes, with the larger one of diameter L and the smaller of diameter rL, where r is the
ratio.
It can be observed that the distance between the corresponding curves produced by arrays of square and circular
floes is modest for the lowest wave period considered and
decreases as the period becomes larger. Similar behavior occurs when the ice distribution is changed, with attenuation
only being affected to a small degree when the wave period
is low and less so when the period increases. However, we
do note that attenuation by arrays of circular floes is consistently smaller than that of square floes of the same area,
which is attributed to less scattering caused by a smoother
circular obstacle.
−2
−3
10
10
(a)
(b)
−3
10
8s
−4
10
8s
−4
10
α
−5
10
12 s
−5
10
−6
10
12 s
−6
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ratio
Figure 7. Comparison of the scattering caused by circular floes (left) and square floes (right) when used in
two-rows arrays. The figures show ice-floe displacement
and water-surface displacement for a obliquely incident
(χ = π/6) wave of period 6 s . Although the low wave
period distinguishes the relative wave fields, when the
number of rows is increased the resulting attenuation coefficients are very similar (0.0378 for an array circles and
0.0535 for an array of squares).
In figure 9 we compare the attenuation caused by arrays
of circular and square floes further and simultaneously investigate the influence of the distribution of bodies. For
this purpose we consider arrays of modules of two circular
or square bodies of different sizes in which the total area
covered by ice in each module is constant (a module for
the circular case is depicted in figure 8). The figure shows
the attenuation coefficient as a function of the ratio of the
16 s
16 s
−7
10
0.8
0.9
−7
1
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ratio
Figure 9. The attenuation coefficient as a function of
floe-length ratio when the array is composed of modules
containing two floes. Results are shown for three different
wave periods and for both square floes (solid lines) and
circular floes (broken). Part (a) corresponds to a higher
concentration of larger floes while the part (b) is for a
lower concentration of smaller floes.
The results presented in this section are representative of
the wide range of tests that we have performed. It is therefore concluded that the influence of floe shape and the distribution of the ice cover on attenuation is modest, particularly
for longer incident waves. For this reason our investigation
will continue using arrays composed of evenly distributed
circular floes, although the authors recognize that this topic
will need further investigation in the future as models become more sophisticated.
3.4. Concentration
The influence of the ice pack concentration in determining attenuation is important and, to be dealt with accurately, requires a three-dimensional model. It has already
X-8
BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
been noted in §2.2 that the non-dimensional attenuation coefficient α is insensitive to the separation of the rows in the
array. Therefore, we choose to calculate a concentration, c
say, in our arrays based on a cell that contains a single floe,
and we write
ported waves changes in the 8 s case and, in this instance,
two linear regimes are identified with similar gradients.
−3
6
A
c = 2,
p
(4)
α
x 10
(a)
4
2
0
where A is the area of the fluid surface occupied by the floe
and p is the periodicity length of the array.
The value of c can be influenced in two ways – through
the size of the floes and via the in-row spacing used. In
figure 10 we examine the way in which in-row separation
affects attenuation. The floes that form the arrays here are
circular, with 25 m radius, 0.5 m thickness and include draft.
As with previous cases the subfigures are for periods 8 s, 12 s
and 16 s.
−4
x 10
α
(b)
1
0
−5
2
α
x 10
(c)
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Concentration
−2
10
α
(a)
−3
10
−4
10
Figure 11. The attenuation shown in figure 10 but
as a function of concentration, c, given in equation (4).
Straight-line fits are also shown (by thick lines) as described in the text.
−3
10
α
(b)
From figure 10 we may thus deduce that
−4
10
α ≈ α0 + α1 c,
−5
10
−4
10
α
(c)
−5
10
−6
10
0
10
20
30
40
50
60
70
80
90
100
In-row separation (m)
Figure 10. The attenuation coefficient as a function of
in-row separation for period (a) 8 s, (b) 12 s and (c) 16 s.
The floes have radius 25 m and thickness 0.5 m.
All three curves shown in this figure demonstrate the intrinsic property of decreasing attenuation with increasing
in-row separation. Over the 100 m interval used the difference in attenuation is approximately an order of magnitude
for all three periods, which implies that the behavior of the
attenuation as a function of concentration is only weakly related to period. The curves display a slight, yet discernible,
upward concavity, noting that α is plotted on a logarithmic
scale. The qualitative nature of the curves is similar in all
cases except for a small interval around 40 m for the 8 s period. At this point the kink in the curve can be attributed
to the occurrence of a change in the number of waves supported by the array. This issue is an aspect of the periodicity
that is required in order to facilitate our solution methods
and, although only producing small anomalies such as the
one shown here, as mentioned previously, it is something
that will provide stimulus for the future development of our
model.
The results presented in figure 10 are also presented in
figure 11 but with attenuation as a function of concentration, c, given in equation 4, rather than in-row separation.
The data sets in this case are also overlaid with least-square
straight-line fits, which demonstrates the linear dependence
of attenuation on concentration and we note the linear scale
used for α in these plots. This linear relationship is interrupted only over the interval in which the number of sup-
where the quantities αj (j = 0, 1) are dependent on the
properties of the floe and the incident wave only. It follows
that the scaling of the attenuation is inversely proportional
to the square of the in-row separation.
Concentration can also be varied through the area of the
fluid surface occupied by each floe. As we have already
demonstrated a linear dependence of the attenuation on concentration, we will isolate its dependence on variations to
the size of the floes by keeping concentration fixed. In a
two-dimensional setting, Kohout and Meylan [2008] found
that beyond a certain, period-dependent value, floe length
does not affect the attenuation in the model. This feature
is related to the ability of a wave to induce the elastic response of a floe only when the ratio of the floe length to the
ice-coupled wavelength is sufficiently large. Once this limit
is obtained any further increment to the floe length will be
irrelevant, as waves are assumed to propagate without energy loss through regions of uniform ice cover. Conversely,
relatively small floes act like rigid bodies. It is not immediately apparent that the simple behavior shown in the twodimensional model will be replicated in three-dimensions,
where motions are not restricted to a single plane.
In figure 12 the dependence of the attenuation on the
horizontal extent of the floes that make up the array is explored. The floes used are circular and we therefore show α
as a function of their radius and set the in-row separation
to be twenty percent of this in order to maintain a constant
concentration. The periods 8 s, 12 s and 16 s are shown and
in all cases the thicknesses of the floes are varied normally
around the mean value of 1 m in order to eliminate length
resonance.
X-9
BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
−1
10
α
(a)
mic scale used here. As usual, the floes used in the arrays
are circular and Archimedean draft is included.
−2
10
−3
10
−1
10
−2
10
α
(b)
−3
10
−2
10
−4
10
−3
10
α
(c)
α
−3
10
−4
10
−5
10
10
−4
20
30
40
50
60
70
80
90
100
10
Radius (m)
Figure 12. The attenuation coefficient as a function of
floe radius for period (a) 8 s, (b) 12 s and (c) 16 s. The
floes have mean thickness 1 m and the in-row separation
is twenty percent of the radius so that a constant concentration is maintained.
As would be expected for the three chosen periods, we see
that for relatively small floes attenuation becomes greater as
radius increases. This relationship becomes less important
as the floes get larger until, at a radius of approximately 50–
70 m, the attenuation is no longer affected by floe size variations. Our three-dimensional model therefore mirrors the
behavior of a two-dimensional model in this respect. There
is also evidence here that the size of the floes required to
attain a stable attenuation coefficient increases as period
increases, which concurs with our explanation of this property being governed by a relationship between floe length
and wavelength. However, for 8 s, 12 s and 16 s incident
waves and a 1 m ice thickness, the ice-coupled wavelengths
are 125 m, 226 m and 394 m respectively, and it is clear that
the limit is not given by a simple ratio.
3.5. Comparison of the three-dimensional and twodimensional models
Having made the above investigations into the dependence of the attenuation coefficient on the key parameters
of the model, we now wish to compare the attenuation rates
predicted by our new three-dimensional model with the existing two-dimensional model of Kohout and Meylan [2008].
In Kohout and Meylan [2008] the attenuation coefficient is
plotted against period for a number of mean thicknesses, as
these were found to be its two principal determinants when
the floes are sufficiently long. From these data the three sets
with thicknesses 0.5 m, 1 m and 2.5 m have been selected for
comparison and these are displayed in figure 13.
These results are shown alongside attenuation coefficients
for our three-dimensional arrays with corresponding ice
thicknesses. For consistency, and following on from our investigation of the dependence of attenuation on floe size in
the previous section, we use floes that are large enough that
the attenuation coefficient has become insensitive to any further extension. We note that, as the flexural rigidity of ice
is strongly associated with its thickness, the size of the floes
required to ensure a settled attenuation coefficient grows as
their thickness increases. The concentration of the arrays is
set as c = 0.5, although, as it results in only linear variations, this parameter is of little consequence on the logarith-
6
7
8
9
10
11
12
13
14
15
16
Period (s)
Figure 13. A comparison of the attenuation predicted
by our three-dimensional model (lines) with that of the
two-dimensional model of Kohout and Meylan [2008]
(symbols) for large floes. Results for the three thicknesses D =0.5 m (solid lines and crosses), 1 m (dashed
lines and circles) and 2.5 m (dot-dash lines and pluses)
are given.
It is clear from figure 13 that the models match reasonably well in a quantitative sense, with the corresponding
attenuation coefficients never differing by more than an order of magnitude. As the two-dimensional model assumes
homogeneity in one spatial dimension and therefore does not
account for any open water between floes in this direction,
it could be anticipated that it would cause more attenuation
than the three-dimensional model. However, this comparison shows that the relationship between the two models is
far more complicated, with the two-dimensional model producing greater attenuation for low periods and the threedimensional model likewise for high periods. This is an interesting finding, as a consistent overprediction of attenuation for low periods and underprediction of attenuation for
high periods has been observed when using two-dimensional
models. We also note that the point at which these regimes
interchange occurs at a higher period as thickness increases.
Qualitatively, although the shape of the curves in the
2.5 m ice thickness case are alike, for the two thinner thicknesses the attenuations predicted by the two models display
distinct differences from one another. Whereas the attenuation coefficients of the two-dimensional model have prominent variations in curvature, on this logarithmic scale our
attenuation coefficients are far straighter. The former may
be a product of the transition between mass and flexure
dominating the scattering process [Vaughan et al., 2007],
which is likely to be present for the 2.5 m thickness also
but occurring for periods out of the chosen range. However,
this feature is not present in the three-dimensional model
where the ice cover consists of an infinite number of individual floes. The attenuation curves predicted by the threedimensional model thus resemble those recently calculated
by Squire et al. [2009] for attenuation in long transects sampled from submarine voyages in the Arctic Basin. We note
though that the curves given here are generally not linear
and display a slight concavity, which changes from upward
for the thinner floes to downward for the thicker floes.
4. Comparisons with experimental data
Unfortunately, within the field of sea ice research, there
are very few high quality data sets that include measurements relating to the evolution of waves in the MIZ and
X - 10
BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
are supported by robust contextual observations such as the
physical properties of the sea ice and meteorological and
oceanographical data. Indeed, at the time of writing, the
best available data were recorded thirty years ago. So, with
the adaptations the polar seas are currently experiencing
due to climate change and the need to improve coupled models by properly assimilating the effects of wave-ice interactions, it is imperative that fresh efforts are made to perform
experiments that will complement theoretical advances of
the kind we have undertaken to create a synergy that progresses our understanding of this complicated phenomenon.
In this context, a voyage has been scheduled for spring 2011
by the Australian Antarctic Division, which will, amongst
other things, carry a group of researchers whose primary
goal is to extract measurements of wave attenuation in icecovered waters.
Notwithstanding the dearth of suitable data and despite
the limitations of the accompanying descriptions of the ice
cover, it is still possible to gain some confidence about the
performance of our model by comparing its predictions with
the historical data referred to above, notably Squire and
Moore [1980] and Wadhams et al. [1988].
4.1. Bering Sea
The MIZ wave attenuation experiment performed during a research cruise by NOAA Ship Surveyor in the Bering
Sea during March 1979 [Squire and Moore, 1980] is, in all
likelihood, the most complete single experiment of its kind
done to date. While a wave buoy recorded off the ice edge,
measurements were made at eight stations aligned with the
principal swell and stretching almost seventy kilometers into
the ice pack. These data were processed to extract attenuation coefficients for five central wave periods. During the
helicopter flights to and from the recording sites within the
pack ice, ice concentration was estimated to be approximately fifty percent and a zonal morphology, edge, transition and interior, that delineates the approximate diameter
of the floes was noted.
The attenuation coefficient that is usually calculated in
practice, a say, is dimensional and such that
E ∝ e−aL
model reported herein. In keeping with the report of the ice
conditions given by Squire and Moore [1980], we use c = 0.5
as the ice concentration in our model and set the average
thickness as D = 0.5 m. To replicate the structure of the ice
cover, three model runs were made, in which the diameters
of floes were 10 m, 25 m and 100 m, these being the mean
values across each zone, and the value of the attenuation
determined for each zone was then used in a weighted average according to the proportion of the ice pack occupied by
the relevant band.
Given that the authors have had to idealize the
formidable complexity and heterogeneity of a real MIZ and
to interpret average parameters to describe the physical
properties of the ice cover, the similarity between the experimental and model attenuation coefficients is most pleasing.
In particular we note that for all but the highest period, the
attenuation coefficients are of the right order of magnitude.
Better still is the agreement to the first significant figure for
the 7.6 s and 6.4 s periods.
As mentioned previously, the underprediction of attenuation for long waves in models that only accommodate scattering is well known. Such behavior should not be surprising
though, as the role of scattering is subordinate to dissipative
processes in this regime [see Vaughan et al., 2009]. For this
reason, the addition of viscosity to parameterize energy dissipation in the model, arising in both the water and, because
of inelasticity, in the sea ice, is expected to improve its accuracy but this will be the topic of a future work. The overprediction of attenuation for short waves has also been observed previously, although it is less well understood. It may
be due to unmodeled effects such as the non-linear transfer
of energy between frequencies, wave generation within the
ice, or resonances, caused by wave-ice coupling, which create new waves that are not fully captured by the model. For
the first time, the authors believe that the sophistication of
the reported model will allow these more esoteric effects to
be fully investigated, now that a robust and accurate threedimensional characterization of MIZ scattering is in hand.
Despite the presence of these foci for future model development, we are encouraged by the concurrence of theory and
field experiment in the band of relevant periods, remembering that the model is parsimonious to the extent of any
propitious parameterizations.
where L is the distance (in meters) into the ice pack. Consequently, to make a comparison with experimental data we
10
must first scale our non-dimensional attenuation coefficient
α so that it describes attenuation per meter. To do this, we
use the definition of the ice concentration c given in equation
(4) and set
10
r
c
,
a=α
A
recalling that A is the area of the fluid surface occupied by a (m−1 )10
a single floe.
Table 1 compares the experimental attenuation coefficients with attenuation coefficients calculated using the
−3
5.5 s
−4
7.6 s
−5
12.2 s
−6
10
Table 1. Comparison of the (dimensional) attenuation coef-
ficients a (m−1 ) calculated during an expedition in the Bering
Sea in 1979 [Squire and Moore, 1980] against those found from
a weighted average of results given by the three-dimensional
model, as described in the text.
Period (s)
12.2
9.4
7.6
6.4
5.5
10−4
Experimental a ×
0.272 ± 0.054
0.438 ± 0.036
0.855 ± 0.049
1.087 ± 0.037
1.214 ± 0.192
Model a ×
0.030
0.201
0.813
1.567
3.237
10−4
−7
10
0
10
20
30
40
50
60
70
Distance (km)
Figure 14. The (dimensional) attenuation coefficient a
as a function of distance into the MIZ predicted by our
model, where the properties of the ice pack derive from
data reported by Squire and Moore [1980].
In Squire and Moore [1980] a curve is given that shows
how the diameter of an archetypal floe at a given penetration
BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
into the ice pack increases from 10 m at the ice edge to just
over 100 m at 70 km. Using this distribution in our model, in
figure 14 we have produced the attenuation coefficient as a
function of penetration for three wave periods. It is interesting to observe the differing behaviors of attenuation at the
three periods considered. Only in the 7.6-s-period case does
the attenuation clearly display the sharp jump at around
35 km into the ice pack where the floe radius rapidly increases. In comparison, attenuation for a 12.2 s wave grows
steadily to begin with, before beginning to level off, and
for a 5.5 s wave the attenuation appears to be insensitive to
distance from the ice margin.
4.2. Kong Oscars Fjord
During September 1979 the attenuation of wave energy
was also measured in a fjord abutting the Greenland Sea.
Conditions were more challenging here than in the Bering
Sea, mainly because all the experiments had to be done from
a land base at Mestersvig, a military outpost with a 1,800 m
gravel runway located in Scoresby Land on the southern
shore of Kong Oscars Fjord in Northeast Greenland National Park. This limited the measurements to within the
fjord, which is not an ideal setting for monitoring the interaction of ocean waves with sea ice as factors such as reflections from the walls of the fjord, refraction, and a limited
range of incidence angles (advantageous in some cases) influence experimental design. Despite these setbacks, full sets
of data from two separate experiments were made and the
results are published in Wadhams et al. [1988] along with an
account of the prevailing ice conditions in Overgaard et al.
[1983]. While suffering from the complications noted above,
these data are regarded as a reliable addition to the Bering
Sea measurements for comparison with the model.
Of the two experiments that took place on the 4th and
10th September respectively, we select the former here as
it is accompanied by a more comprehensive description of
the prevailing ice conditions. The ice concentration was estimated at 0.3 and the diameters of the floes were in the
range 50–80 m. It was not possible to estimate a mean ice
thickness and we follow Kohout and Meylan [2008] in using the value given in the same location the previous year,
which was 3.1 m. These ice properties are significantly different from those in which the Bering Sea experiment was
conducted, so the experiment provides a valuable second test
case.
Table 2 compares the experimental attenuation coefficients from the 4th September in Kong Oscars Fjord with
attenuation coefficients calculated using our model, with
geometrical parameters intended to replicate, as best we
can, the conditions reported during the experiment. Specifically, the floe diameter was set as 65 m, the concentration
at c = 0.3 and the ice thickness was given a Gaussian distribution around the mean value 3.1 m and standard deviation
0.5 m.
Again, we note that there is a general agreement, to at
least the order of magnitude, between the experiment and
Table 2.
Comparison of the (dimensional) attenuation
coefficients a (m−1 ) calculated during an expedition in the
Greenland Sea on 4th September 1979 [Wadhams et al., 1988]
against those calculated from our three-dimensional model.
Period (s)
14.03
11.88
10.31
9.10
8.14
Experimental a × 10−4
0.29 ± 0.27
0.73 ± 0.25
1.23 ± 0.19
2.01 ± 0.17
2.66 ± 0.22
Model a × 10−4
0.11
0.30
0.53
0.84
2.24
X - 11
model. However, it is evident in this case that the model
underpredicts the attenuation when compared to the experiment and, in the four highest periods, this underprediction
is approximately a factor of 2.4–2.65, becoming closer to the
experimental value as period decreases. It is important to
note though, that these errors may not be related to the accuracy of the model but rather to deficiencies in the reported
experimental conditions, as noted earlier, or to an insufficient description of the ice state. Despite these hindrances
the similarities between the experiment and the model are
reassuring.
The experimental and model attenuation coefficients are
extremely well matched for the lowest period. In regard to
our earlier investigation, it is clear that the rapid change
in the attenuation coefficient between 9.1 s and 8.14 s can
be attributed to the addition of extra traveling waves at a
nearby period and our previous comments concerning this
phenomenon apply. It is important not to become injudicious about the similarity of the attenuation predicted here,
however, and we note that this indicates once more that the
model attenuation coefficient is more sensitive to wave period than experimental data would suggest. On the other
hand, we must not dismiss the fact that we have another
example of good agreement between the model and experimental data over a key range of periods.
5. Summary
In this work we have used a model, constructed to represent the passage of ocean waves through the MIZ, to study
the rate of wave attenuation with respect to the properties of the ice cover and the incident wave field. A threedimensional description was employed, which treats the MIZ
as a vast collection of separate floating elastic floes whose
motions are interrelated. The model is based on the assumption of linear motions and wave attenuation is caused only
by the scattering produced by the floes. Flexure of the ice
cover is identified as the main component of the propagation
of wave energy and other non-linear effects such as inelasticity, turbulence and inter-floe collisions are neglected.
During the numerical investigation the following discoveries were made. Firstly, it was demonstrated that the attenuation coefficient has a roughly exponential dependence on
the angle at which the incident ocean wave impacts on the
array of floes, with oblique waves diminishing fastest. This
provided an indication of some slight collimation, although
at lower periods a contrasting spreading of the directional
wave field was evident. Next, it was shown that the addition
of an Archimedean draft has an insubstantial effect on attenuation. Similarly, changing the shape and distribution of
the floes was found to have little bearing on the attenuation
produced by a large array. The attenuation predicted by the
model was established as being directly proportional to the
concentration of the ice cover. Furthermore, we studied how
attenuation behaves in relation to the two determinants of
concentration – floe spacing and floe diameter. It was noted
that there is a limit at which attenuation becomes insensitive to the latter but that for meaningful dimensions its
value can differ by an order of magnitude according to the
floe size present.
Comparisons were also made of the three-dimensional
model to an existing two-dimensional model and to two sets
of experimental data of wave attenuation in the MIZ. For
the former, it was clear that the three-dimensional model,
in accordance with the two-dimensional model, is highly dependent on the average value of ice thickness and on wave
period. Although, in the examples shown, the attenuation coefficients predicted by the two models did not differ
by more than an order of magnitude, their qualitative appearance was distinct. In particular, the attenuation coefficients produced by the three-dimensional model were greater
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BENNETTS ET AL.: WAVE ATTENUATION IN THE MIZ
than the two-dimensional model for low periods and were
smaller for high periods. Both comparisons with experimental data gave positive agreement and provided confidence in
the model’s ability to describe attenuation due to scattering.
However, the familiar elements of underprediction for high
periods and overprediction for low periods were still present.
As mentioned in the introduction, we believe that the
model providing the basis for this work contains the potential to be assimilated into an oceanic general circulation
model. Therein, our model would interact with embedded
ice rheology and thermodynamic codes to provide information and predictions about the distribution of ocean wave
activity in the sea ice and associated effects, such as waveinduced strain. The latter will contribute to the morphological evolution of the ice pack along with deformations arising because of currents and winds acting through the large
scale sea ice constitutive equation, e.g. viscous-plastic rheology. However, despite the encouraging performance that
the model has shown in this work over meaningful ranges of
periods, it is clear that opportunities still exist for its improvement. For instance, in terms of the interaction theory,
we wish to eliminate the periodicity restraint, so that the
rapid change in the attenuation coefficient over the intervals
where additional waves cut-in will be eliminated. Furthermore, methods must be developed to incorporate some nonscattering mechanisms, such as dissipation, into the model
that will improve its accuracy in the high and low period
regimes.
As a final note we re-iterate that there is the promise
of new data to complement our theoretical research in the
near future. This is an exciting prospect and will certainly
encourage further research to help replicate the physical situation as accurately as possible.
Acknowledgments. Supported by the Marsden Fund Council from Government funding administered by the Royal Society
of New Zealand. The authors also thank Alison Kohout for providing data for comparison.
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